Find the area between the cycloid : x=a (t-sint), y=a (1-cost) [0

x=a (t-sint)

y=a (1-cost)

the area, as usual, will be

∫ydx

but notice that

dx / dt = a (1-cost) →dx=a (1-cost) dt

so the integral becomes

∫ydx = ∫ 2π (upper integral limit) 0 (lower integral limit) a^2 (1-cost) ^2 d t